Transitive graphs uniquely determined by their local structure

TL;DR

This paper proves that certain transitive graphs, exemplified by the grandfather graph, are uniquely determined by their local structure, meaning large enough finite subgraphs can uniquely extend to the entire graph.

Contribution

It establishes a uniqueness property for transitive graphs based on local subgraph structures, specifically demonstrating this for the grandfather graph.

Findings

The grandfather graph is uniquely determined by its local structure.

Large enough finite subgraphs can be extended uniquely to the entire transitive graph.

The result characterizes the extent to which local structure determines global graph properties.

Abstract

We show that the "grandfather graph" has the following property: it is the unique completion to a transitive graph of a large enough finite subgraph of itself.